I'm not sure they were ever replaced; the terms co-exist. I believe the terms all stem from the general concept of writing a name on a piece of paper.

To "register" your interest in an something, you might sign a piece of paper, or instruct a person to "sign [you] up".

To show that you entered a physical event, you would sign an entry log on a piece of paper, then having "logged in" or "signed in", and then do the opposite on your way out.

Similarly you might refer to reproducing your physical signature as having "authenticated" yourself, if there's something to compare it to.

What makes one particular set of terms the right set given the apparent origins?